A Model for Soft Interactions based on the CGC/Saturation Approach and the BFKL Pomeron

Strong interactions at high energy are considered a typical example of processes that occur at long distances, where the unknown mechanism which confines quarks and gluons, plays a confusing role in undermining all efforts to treat these process theoretically. Consequently, the description of these soft processes are usually made using high energy phenomenology, based on Pomeron-Reggeon calculus [1]. To ameliorate this difficulty, the Tel Aviv group (Gotsman, Levin and Maor) has suggested and formalized an approach, which attempts to describe soft interactions by adopting well established aspects of QCD in the saturation region [2], and extending these to the region of soft interactions (low Q). Our basic assumption, is that the BFKL Pomeron [3], at high energies, describes both hard and soft scattering. The processes that our model has been successfully applied are proton-proton scattering over the energy range 546 GeV ≤W ≤ 57 TeV: include total, elastic and diffractive cross sections as well as the forward slope [4]: inclusive production [5]:long range correlations [6]: and the survival probability of central exclusive production [7]. To illustrate our approach, I will limit myself and only discuss our results, and compare these to the relevent experimental data for the proton-proton total, elastic and differential cross sections, as well as inclusive production.


Introduction
Strong interactions at high energy are considered a typical example of processes that occur at long distances, where the unknown mechanism which confines quarks and gluons, plays a confusing role in undermining all efforts to treat these process theoretically.Consequently, the description of these soft processes are usually made using high energy phenomenology, based on Pomeron-Reggeon calculus [1].
To ameliorate this difficulty, the Tel Aviv group (Gotsman, Levin and Maor) has suggested and formalized an approach, which attempts to describe soft interactions by adopting well established aspects of QCD in the saturation region [2], and extending these to the region of soft interactions (low Q 2 ).Our basic assumption, is that the BFKL Pomeron [3], at high energies, describes both hard and soft scattering.
The processes that our model has been successfully applied are proton-proton scattering over the energy range 546 GeV ≤ W ≤ 57 TeV: include total, elastic and diffractive cross sections as well as the forward slope [4]: inclusive production [5]:long range correlations [6]: and the survival probability of central exclusive production [7].
To illustrate our approach, I will limit myself and only discuss our results, and compare these to the relevent experimental data for the proton-proton total, elastic and differential cross sections, as well as inclusive production.

Phenomenological Input
One of the most well established principles of high energy scattering is that the resulting cross section should not violate the Froissart bound (F.B.).
There is a very simple criterion due to Heisenberg (quoted by Kovner and Wiedemann [9]) relating the impact parameter dependence of the profile of the matter density in any target, and the violation of the F.B.
(i) If the matter density decays exponentially at the periphery i.e. ρ(b) ∝ e −mb , then for the production of the lightest particle of mass m, (E ∼ s m ), the target energy density is Eρ(b).Scattering can only take place for b ≤ b max .Since this satisfies the F.B.
(ii) Conversely, if the density distribution in the target is power-like, i.e. ρ(b) ∝ b −λ .Hence, i.e. σ tot ∝ a power of s, which leads to a violation of the F.B. In building our model, we need to overcome a deficiency i.e. the fact that the BFKL Pomeron does not lead to shrinkage of the diffractive peak, and has no slope for the Pomeron trajectory.This can be cured by introducing a non-perturbative correction at large impact parameter, which also assures satisfying the Froissart-Martin bound for σ tot .
Consequently, in our model we fix the large b behaviour by assuming that the saturation momentum Q 2 s , has the following form: The parameter λ = ᾱS χ (γ cr ) /(1 − γ cr ), in leading order of perturbative QCD (λ = 0.2 to 0.3).
The parameter m which is introduced to describe the large b behaviour, and determines the typical sizes of dipoles inside the hadron.
Our model also includes two additional scales m 1 and m 2 , which describe two typical sizes in the proton wave function.One can associated these with: (i) the distance between the constituent quarks; size of the proton R p ≈ 1 m 1 .and (ii) m 2 can be associated with the size of the constituent quark; R q ≈ 1 m 2 .Altinoluk et al [10] have proved the equivalence of the CGC/saturation approach and the BFKL Pomeron calculus for a wide range of rapidities , Y ≤ 2 ∆ BFKL ln

Dressed Pomeron in MPSI approximation
For a description of the process see Fig. 1.
∆ BFKL , only large Pomeron loops with rapidity O(Y) contribute at high energies → can sum such loops using MPSI approximation.
For the BFKL Pomeron λ = 4.88 ᾱs while ∆ BFKL = 4ln2 ᾱs ≈ 0.2.The resulting Green function of the Dressed Pomeron is given by: The BFKL amplitude in the vicinity of the saturation scale is denoted by , a = 0.65, the critical anamolous dimension is given by γ cr ≈ 0.37.

Parameters of the Model
We used two inputs: r = R and  The opacity is given by: , where Ḡdressed (Y; b ′′ ) .The factor 1.29 originates from estimates of the 3I P vertex in the CGC/saturation approach.

Basic formalism for the two channel model
Following Good-Walker [11] the observed physical hadronic and diffractive states are written where Functions ψ 1 and ψ 2 form a complete set of orthogonal functions {ψ i } which diagonalize the interaction matrix The unitarity constraints can be written as At high energies a simple solution to this equation is ) denotes the contribution of all non-diffractive inelastic processes.

Physical observables for elastic and low mass diffraction
elastic amplitude : elastic observables : 'GW' denotes the Good -Walker component, that is responsible for diffraction in the small mass region.
The parameters of our best fit are given in  Comparison of model results with experimental data: the energy behaviour of the total (Fig. 2-a), inelastic (Fig. 2-b), elastic cross section (Fig. 2-c), as well as the elastic slope B el (Fig. 2-d), and single diffraction (Fig. 2-e) and double diffraction (Fig. 2-f) cross sections.The solid lines show our fit.The data has been taken from Ref. [12] for energies less than the LHC energy.At the LHC energy for total and elastic cross sections we use [13] data and for the single and double diffraction cross sections are taken from ref. [14].The dotted line in Fig. 2-f) which is obtained assuming factorization is discussed in [4].

Application of the CGC/saturation approach to Inclusive Production
See [5] for details.
Inclusive production occurs in two stages: First stage: Production of a mini-jet with typical transverse momentum Q s : Q s (saturation scale)≫ soft scale.Second stage: Decay of minijets into hadrons, which is treated phenomenologically.
For mini-jet production we use the k T factorization formula: where φ h i G denotes the probability to find a gluon that carries the fraction x i of energy with k ⊥ transverse momentum.and ᾱS = α s N c /π, with the number of colours equal to N c .
Where 1  2 Y+y = ln(1/x 1 ) and 1 2 Y−y = ln(1/x 2 ).φ h i G is the solution of the Balitsky-Kovchegov(BK) non-linear evolution equation, and can be viewed as the sum of 'fan' diagrams of the BFKL Pomeron interactions, shown in Fig- 3.
For the sake of simplicity all other indices in φ (x 1 , p T − k T ) and φ (x 2 , k T ) are omitted.Eqn.(1) can be rewritten as a Mueller diagram ( Fig- 3b), and the inclusive cross section is given by: where GI P (y) = φ 0 exp (λ (1 − γ cr ) y) and N BK is an approximation to the numerical solution of the BK equation.The mass of mini jet is given by m 2 jet = 2m so f t p T .Since the typical transverse momentum is equal to the saturation scale, we have Values of parameters have been extracted from the diffractive and elastic data.The only free parameters are a I PI P and r 2 0 .Our curves are calculated for a I PI P = 0.21 and r 2 0 = 8, which have been determined from the experimental data.

Our Model Results for Inclusive Production
The results of our fit and the relevent experimental data are displayed in Fig ( 4) and Fig. (5).

Figure 1 .
Figure 1.a) Dressed Pomeron in MPSI approximation and b) Sum of net diagrams.Wavy lines describe BFKL Pomerons.The grey blobs stand for triple Pomeron vertices, while black blobs show the hadron-Pomeron vertex g(b).

R 2 )
S (b) exp (λ Y).The phenomenological profile function S (b) = m 2 2π e −m b with normalization d 2 b S (b) = 1.We need to introduce four constants: g i and m i (i = 1, 2), to describe the vertices of the hadron-Pomeron interaction g i (b) = g i S I P (b) with S I P (b) = m 3 i b 4π K 1 (m i b) ; S I P (b) Fourier image −−−−−−−−−−−−−→ (

Figure 2 .
Figure2.Comparison of model results with experimental data: the energy behaviour of the total (Fig.2-a), inelastic (Fig.2-b), elastic cross section (Fig.2-c), as well as the elastic slope B el (Fig.2-d), and single diffraction (Fig.2-e) and double diffraction (Fig.2-f) cross sections.The solid lines show our fit.The data has been taken from Ref.[12] for energies less than the LHC energy.At the LHC energy for total and elastic cross sections we use[13] data and for the single and double diffraction cross sections are taken from ref.[14].The dotted line in Fig.2-f) which is obtained assuming factorization is discussed in[4].

Figure 3 .
Figure 3.The graphical representation of Eqn.(1) (Fig-3a).Wavy lines denote the BFKL Pomerons, while the helical lines illustrate the gluons.In Fig-3b the Mueller diagram for inclusive production is shown.

Table 1
, while our results and comparison with the relevant experimental data are displayed in Fig.2.